Gear tooth



May, 1923. L

l J. M. LABBERTON GEAR TOOTH Filed Nov. 6, 1919 WITNESSES: NVENTOR fhfymmmfm .i rara er :aan ra'.

:ionen aanname a vente".

PENNYSLVANIA, ASSIGNOR T WEYS'I'ING- GEAR Toorn.

Application filed November 6, 1919. Serial No. 336,069.

ot l.ll`ll :insl)tirg, in the county of Allegheny and State ot l'lennsylvania, have' invented a ner: and useful lmprorement in Gear Teeth. v

of which the following is a specification.

My invention relates to gear teeth as ena-y Yployed on gears and racks and it has'tor its ohjeet to provide involute teeth that Shall 'oe strongifr than those commonly employed at present and that shall further mesh with adequate clearance, with a substantially unitorni percentage ot rolling and with au optiinun'r pressure angle.y irrespective ot the ratio employed.

`lleretot'one. a. constant pressure angle has been used in the design of gear teeth ot the involute form over the entire range ot gear ratios and of gear Wheel diameters. In the ease of a pinion having a relatively small number of teeth meshing .vith a `gear wheel having a large numhrr of teeth. the form ot' the teeth in hoth `wheels must he modified in order to permit the gear and the pinion to operate properly without interferenre. rThe amount ot rolling between ril-operating gear te th will with the gear ratio used. with the result that a gear or a pinion Vwhich may operate properly and with rela tively small wear in one application \vill lio suhjented to inurh greater wear in another -fipplir-ation. The addendum and the dedendum of the gear teeth have heen made suhstaat-hilly equal save tor a .small amount ot rlearanrr at 'the root ot the tooth. resulting in the thirlmoss oil the teeth of hoth the pinion and the gear at the pitch rircle lining the .saine and. ronsiquenlly, of relatively dilli'ri-nt strengths.

.ln praetieing my invention. l provide a. moditad form ot ini'olute tooth tor ro-operating gear wheels, hased upon a mathematif'al analysis ot the oleYnents entering into the problem ol meshing gear teeth. ving enne. made this analysis and having derived. several relatively simple formula. l ..in ahle. their use. to r'alenlate the proper Constants to he used in laying ont the tooth forni.. The data usually available whenlayform of teeth for a set ot eogear Wheels includes the gear gear Center distant-e and the deahove-n'ientioned formulae embodying Ymy method, I calculate the pressure angle, the addendum and the dedendum of the teeth and the relative tooth thickness at the pitch circle. and may then proceed to lay out'the tooth form for the particular application.

.Referring to the drawings` Figure 1. is a diagram showing two co-operating teeth of a spur pinion and gear, together With certain other details which will he used in the mathematical analysis of the problem. and Fig. 2f is a Curve showing the ratio of the thicknessesot the teeth ot the :ttl and #2 gear Wheels at the pitch oirrle as a function ot' the gear ratio.

Turning now to the specific details of my invention` Let 'rlzradius of pitoh circle of the #l gear wheel. rgrzradins of pitch (-irrle gear Wheel. r,:radins of hase Circle of the :fi-l

gna r wheel] of the #e Zzradius ot hase circle of the #Q gear s. represent. the renter ot the #l gear wheel. f3.. represent the' contri-vol' the #Q .gear

wheel. 2,2.. represent the line joining the gear renters.

n represent. the pointal. whirl) the line .e122 intersvrts th(` pitrh rirrlos of the two gear wheelsv my rol'iresent the lino of pressure ofthe, co-operating gear teeth drawn as a tangent. to the. two hase circles .fm-x and q1/:1/1.

For the purposes of our analysis` the tw curves S and T, representing the (so-operating gear teeth surfaces, are assumed to pass through the point o.

Draw the lines 21a: and 2 1/ from the res ective gear wheel centers at right angles to t e line my.

The angles 021m and ,0223/ are equal and may be designated by the letter a. rlie involute S will meet its base circle mail at a. point wg, and the involute T will meet its base circle g/g/1 at a point 7/2.

Draw the lines 21a'Q and 22g/2 and by the construction the angle mslm? will be equal to yzazy and may be designated by Y.

Let si represent a point on the curve S at which the tip of the curve l of the romper-v ating tooth will first make Contact under operating conditions. The evolute oiC the involute curve :E251 will be the arc 11a/r2 and We may designate the angle m32, in: by C1. y

Let t1 represent a point on the curve T at which the tip 'of the curve S'of the (co-operating: tooth will last make Contact under operating conditions. The are i/, ?/3 will be the evolute. of the involute curve i/1 and we may designate the angle 1/2223/3 by c2.

The length oi the line .T01-r1 sin a.

The length of the involute Therefore, zw-A- Consider the section S10 of the involute generated by the evolute of the. are @w3 of the angle malais, which is equal to (b-c1) k 2 2 :T @L C12) radians and a portion of the Contact surface T on the cci-operating tooth of the .#2 gear wheel must mate with the section S.- Let this eo-operating surface be that part of T between the parts b and 252.

4ra ily The length of 'the cubperating surface Lgo is, from the drawing rThe proportion of 'the surface T on which rolling of the two Cooperating-tooth surfaces occurs is elo sliding of the. two surfaes occurring; over the remainder thereo't. liet this ratio and 'r2 K: *will 25e-wwwa la Il An inspection of the above equation shows that the ,rolling is a maximum when 6:51, but this is a condition impossible to realize with gear teeth, as S and T 'would then equal Zero and we would have the theoretical eondition of two (3o-operating f'ear 'wheels7 without teeth, i. e., two cylindrical surfaces having frietional driving engagement. The designer ii'iiist therefore decide what amount of rolling is desired or what amount will rive the best operating results, and it may he noted that K will be a number less 'than unity.

lf we represent the ratio by and sut stitute in the above forniulawe get and in a similar mannewe may obtain The perimeter of the pitch circle of the. #l gear wheel=21rr1.

The arc per tooth of the #l gear wheel o Let i21l=7l1=the length of contact sur- 1 face on the tooth face..

This is also equal to The length of contact surface tltz of the tooth from the drawing nu 1 2] g1- 2 2 +2 im@ om 1 Egizia-vg+{Math-mula] It may be noted that the angle of action for theslil gear Wheel and and for the #2 gearwheel =bcz+R(b-c,)

The angle of action for the respective gear Wheels should be not less than the angle subtended by one tooth itch.

The addendum for t e I:ti-2# gear Wheel tooth may be determined by reference to Fig. 1, by solving for the length of line 22152. (not shown).

The1ength of une @aft/MMatbeaijk;

The addendum is est, r2 'Therefore the addendum of #2 gear Wheel tooth lk-21MB@-calwfea c Similarly, the addendum for the #l gear wheel tooth Fig. 2 shows a curve representing the nelabe assumed that it is desired to lay out a.

tooth form for a pair of cti-operating spur gear Wheels having a given distance between gear centers. the diametral` pitch being known, as is also the desired gear ratio. Select a value of K (percentage ot rolling) somewhat-lower than unity. Calculate c, and c2 by using the formul (2) and (3).

Then substitute these values in formula (4) and solve for a the pressure angle, noting that bztan a from formula (l).

Then determine the addenda vtor the respectve gear Wheels, using formulae (5) and (6), and allow the usual clearance at the root of the teeth. By reference to the curve of lFig. 2, the relative thicknesses of the teeth of the tivo gear wheels at the pitch cir cle may be deternfiined.A

We'now have the pressure angle, the addende. for the teeth of the respective gear Wheels, and the tooth thicknesses at the pitch line,'and We` may proceed to lay ont the teeth of the two gear Wheels.

While l have illustrated the use of my' 'Iii method for determining. the dimensions of i the teeth of spur gear Wheels, it may be used also for bevel gearwheels and for helical gear teeth. 1

By the use of the above disclosed method .embodying my invention, lprovide a gear tooth,'the pressure angle of Which-varies not only with the gear ratio but also-with the number of teeth in the respective gear Wheels.' By inspection of the formulae, it may be noted that the pressure angle is larger for the higher gear ratios. which will tend to reduce interference between the cooperating teeth. The location of the points of initial and of final contact between cooperating teeth isfa function of the amount of rolling action and of the gear ratio and,

as the values ot c, and c2 can never be zero, (see Formulee 2 and 3) each of these tivo circle of its own gear, and hence. there can'be no interference. The addenda ot the r spective teeth are different, in order to secure a' better percentage of rolling, the

points will always be outside of the base strength of the teeth in co-operating gear wheels being balanced by the adjustment of the pitch line thickness of the respective teeth. By roportionin the pitch-line thicknesses ci) the teeth o the #l and :#:2

gear wheelsfin accordance with the curve shown in Fig. Q s the teeth in the two gear wheels may be made substantiallyv equal in strength. The rolling action is substantially .the same tor all gear ratios and tor all numsure angle. of a pair ot (1o-operatinggear wheels having inrolute teeth, when the gear center distance, the ,crear ratio and the diametral pitch are known, which con'iprises the Selection ota numerical value for the percentage of rollingi action hetween co-op erating teeth. which. value is less than unity. calculating the values c, and '(2 in terms of 7) from the formulae @E @eiwit wherein K" represents the assumed numerical value of rolling action and R z gear ratio which is equal to or less than unity, substituting the values of el and c2 in the formula Where. l

ggznuinber ot' teeth in larger gear wheel,

l) :tan a. and

fr :pressure angle and then solving for a.

2. The -method ot' determining the pressure angle and addenda of a` pair ot co-0perating gear Wheels having involute teeth, when the gear center distance, the gear ratio and the diametral pitch are known,

Maafeee f which comprises the selection of anumerical value for the A,percentage of rolling action between co-operatin; r teetlnrwhich value is less'than unity. calculating the values c1 and c., in terms 4ol b from the 'formulae noun-1 "ft 'HRK il?) 2. L wg wherein l( represents the assumed nume-rr eal value ot rolling action and R z gear ratio which is equal to or less than unity, substituting the values ot' c1 and 02 in the formula 1r cos (zy i: 2N[{L+it(b-lc,i}2,c,2:| where l(/2:numher ot teeth in larger gear wheel, -b :tan a and rz :pressure angle, and then solvingr for (1,' in calculating the addendum ot' the teeth of the larger gear,

wheel hy t'ormula: addendum of larger gear wheel teeth :Wegman corny-a, wherein 'gzradius of the base circle. of the larger gear wheel and rgzzradius ol' the pitch circle ofthe larga` l gear wheel; and calculating;- the addendum ot' the teeth of the smaller 2 {ear wheel hy formula: ad-

dendum ot' smaller gear wheel teeth i' k12-'T1 wherein /clzradius ot' the base circle of the smaller gear wheel and rlzradius of the pitch circle of the smaller gear wheel. In testimony whereof, I have hereunto subscribed my naine this 3rd day of Nov.

JOHN M. LABBERTUN. 

